Outage Behaviors of NOMA-based Satellite Network over Shadowed-Rician Fading Channels
Xinwei Yue, Yuanwei Liu, Yuanyuan Yao, Tian Li, Xuehua Li, Rongke Liu, Arumugam Nallanathan
aa r X i v : . [ c s . I T ] M a r Outage Behaviors of NOMA-based SatelliteNetwork over Shadowed-Rician Fading Channels
Xinwei Yue,
Member, IEEE,
Yuanwei Liu,
Senior IEEE,
Yuanyuan Yao,
Member, IEEE,
Tian Li,
Member, IEEE,
Xuehua Li,
Member, IEEE,
Rongke Liu,
Senior IEEE, and Arumugam Nallanathan,
Fellow, IEEE
Abstract —This paper investigates the application of non-orthogonal multiple access (NOMA) to satellite communicationnetwork over Shadowed-Rician fading channels. The impact ofimperfect successive interference cancellation (ipSIC) on NOMA-based satellite network is taken into consideration from theperspective of practical scenarios. We first derive new exactexpressions of outage probability for the p -th terrestrial userand provide the corresponding asymptotic analysis results. Thediversity order of zero and p are achieved by the p -th terrestrialuser with ipSIC and perfect successive interference cancellation(pSIC), respectively. Finally, the presented simulation resultsshow that: 1) On the condition of pSIC, the outage behaviors ofNOMA-based satellite network are superior to that of orthogonalmultiple access; 2) With the value of residual interferenceincreasing, the outage performance of terrestrial users with ipSICis becoming worse seriously; and 3) Infrequent light shadowingof Shadowed-Rician fading brings the better outage probabilitycompared to frequent heavy and average shadowing. Index terms—
Non-orthogonal multiple access, satellitecommunications, outage probability, shadowed-rician fadingI. I
NTRODUCTION
With development of Internet of Things (IoT) and satellitecommunication business needs, the IoT-based satellite net-works have received a vast amount of attention, which hasbeen viewed as a crucial application scenarios. Currently, alarge number of satellite machine-type-communication ter-minals on the ground have introduced a great challenge tomultiple access for the satellite communication networks. Non-orthogonal multiple access (NOMA) is an effective approachto meet the requirements of massive connections [1].Until now, the use of NOMA has been confirmed to havebetter performance gain from the perspective of improving thespectral efficiency and user fairness [2, 3]. By extending theconcept of NOMA to cooperative communication, the authorsof [4–6] discussed the users’ outage behaviors by regarding thenearby user as a relay. To present the valuable insights of se-curity performance, the authors in [7–9] have investigated the
X. Yue, Y. Yao and X. Li are with the Key Laboratory of Modern Measure-ment & Control Technology, Ministry of Education and also with the Schoolof Information and Communication Engineering, Beijing Information Scienceand Technology University, Beijing 100101, China. (email: { xinwei.yue,yyyao, lixuehua } @bistu.edu.cn).Y. Liu and A. Nallanathan are with the School of Electronic Engineeringand Computer Science, Queen Mary University of London, London E1 4NS,U.K. (email: { yuanwei.liu, a.nallanathan } @qmul.ac.uk).Tian Li is with the 54th Research Institute of China Electronics TechnologyGroup Corporation, Shijiazhuang Hebei 050081, China and also with BeijingUniversity of Posts and Telecommunications, Beijing 100876, China (email:[email protected]).R. Liu is with the School of Electronic and Information Engineering,Beihang University, Beijing 100191, China (email: rongke [email protected]). secrecy outage probability and maximize the sum secrecy rateof NOMA systems by invoking joint precoding optimization.Furthermore, a pair of NOMA assisted caching strategies wereproposed to emphasize the wireless caching [10]. To betterunderstand NOMA assisted unmaned aerial vehicle networks,the authors of [11, 12] have evaluated the performance in termsof both outage probability and fly trajectory.Satellite networks are expected to be a complementaryrole of terrestrial communication systems, since it is capableof supplying the spacious coverage and short deploymenttime [13, 14]. Hence applying NOMA technology to satellitecommunications will be further a promising way to extendthe applications of the integration between space and earth.To evaluate the performance of NOMA satellite networks, theauthors in [15] researched the outage probability of groundusers, while it did not consider the outage probability per-formance of any one of M users under the condition oforder statistics. From the view of practical scenarios, theSIC procedure exists the potential concrete issues i.e., errorpropagation and complexity scaling, which will result in errorsin decoding process. Hence it is important to take theseundesirable influence from imperfect successive interferencecancellation (ipSIC) into consideration [16]. To the best ofour knowledge, the outage behaviors of terrestrial users withipSIC have not been well evaluated.Triggered by these treatises, we derive new exact andasymptotic expressions of outage probability for the orderedterrestrial users. The impact of channel parameters on NOMA-based satellite network is discussed in detail. Numerical resultscorroborate our analyses that: 1) On the condition of pSIC,the outage behaviors of NOMA-based satellite network aresuperior to that of orthogonal multiple access (OMA); 2) Withthe increasing of residual interference, the outage behaviorsof terrestrial users with ipSIC are becoming worse seriously;and 3) Infrequent light shadowing of Shadowed-Rician fadingresults in a reduced outage probability.II. N ETWORK M ODEL
Consider a NOMA-based satellite communication scenario,where a satellite broadcasts the superposed information toterrestrial users. Assuming that M users randomly distributewithin the coverage of the satellite. The satellite and ter-restrial users are equipped single antenna, respectively. TheShadowed-Rician model is employed to describe satelliteland links. We assume that h p denotes the channel fadingcoefficient from satellite to the p -th terrestrial user. Without loss of generality, the corresponding channel gains betweenthe satellite and M terrestrial users are ordered as | h | ≤| h | ≤ · · · ≤ | h M − | ≤ | h M | [17]. These satellite-userslinks are disturbed by additive white Gaussian noise (AWGN)with mean power ˜ N . In light of the above assumptions, thecumulative distribution function (CDF) and probability densityfunction of channel gains from satellite to the p -th user underthe unordered conditions are given by [18] F | ˆ h p | ( x ) = α p ∞ X k =0 ( m p ) k δ kp ( k !) β k +1 p γ ( k + 1 , xβ p ) , (1)and f | ˆ h p | ( x ) = 12 b p (cid:18) b p m p b p m p + Ω p (cid:19) m p e − x bp × F (cid:18) m p ; 1; x Ω p b p (2 b p m p + Ω p ) (cid:19) , (2)respectively, where γ ( a, x ) = R x t a − e − t dt denotes the lowerincomplete Gamma function [19, Eq. (8.350.1)]. ( m ) k =Γ ( m + k ) / Γ ( m ) is the Pochhammer symbol and Γ ( m ) denotes the Gamma function. δ p = Ω p /2 b p /(2 b p m p + Ω p ) , α p = (2 b p m p /(2 b p m p + Ω p )) m p /2 b p , and β p = 1 / b p . Ω p and b p denotes the average power of the line of sight(LoS) component and multipath component, respectively. m p is the Nakagami- m parameter ranging from zero to infinite. F ( a ; b ; x ) denotes the confluent hypergeometric function[19, Eq. (9.100)].In NOMA-based satellite network, the satellite transmitsthe superposed signals to multiple terrestrial users. Hence thereceived signal y p of the p -th user can be written as y p = h p M X i =1 p η i G s G i ( ϕ i ) a i P s x i + n p , (3)where the power allocation factor a i of the i -th user satisfies M P i =1 a i = 1 and a ≥ a ≥ · · · ≥ a M − ≥ a M to ensurethe users’ fairness. It is worth noting that the optimal powersharing strategy is capable of enhancing the performance of thenetwork, which will be taken into account in the future work.The normalized transmission power at the satellite is denotedby P s . The i -th user’s signal x i is assumed to have zeromean and unit variance and n p ∼ CN (0 , ˜ N ) denotes AWGN.In addition, η i = ( λ /4 πd i ) denotes the free space losscoefficient of one beam. The wavelength is λ = C / f c , where C and f c denote the light speed and frequency, respectively. d i denotes the distance between the satellite and the i -user. G s isthe antenna gain at the satellite. Given the i -th user’s location, ϕ i represents the angle between it and beam center comparedto the satellite. Hence the beam gain G i ( ϕ i ) is given by [20] G i ( ϕ i ) = G i (cid:18) J ( u i )2 u i + 36 J ( u i ) u i (cid:19) , (4)where G i denotes the antenna gain of the i -th user and u i =2 . ϕ j /sin ϕ j . ϕ j is the constant 3-dB angle forthe beam. J ( · ) and J ( · ) denote the first-kind Bessel functionwith order one and three, respectively. Following NOMA procedures [2], the received signal tointerference and noise ratio (SINR) at the p -th user to detectthe information of the q -th user ( p > q ) is given by γ p → q = φ p ρ | h p | a q φ p ρ | h p | M P i = q +1 a i + ηρ | h I | + 1 , (5)where ρ = P s ˜ N denotes the transmit SNR, φ p = η p G s G p ( ϕ p ) and η ∈ [0 , . η = 1 and η = 0 denote ipSIC and pSIC,respectively. Without loss of generality, we assume that h I denotes the residual interference, which follows a complexGaussian distribution with zero mean and variance Ω I i.e., h I ∼ CN (0 , Ω I ) .Then the received SINR of p -th user detect the informationby treating M − p users’ signals as interference is given by γ p = φ p ρ | h p | a p φ p ρ | h p | M P i = p +1 a i + ηρ | h I | + 1 . (6)After the information of M − users can be detected, thereceived SINR for the M -th user is given by γ M = ρa M | h M | φ M ηρ | h I | + 1 . (7)III. PERFORMANCE EVALUATION
1) Outage Probability:
The SIC is carried out at the p -th user by detecting and canceling the i -th user’s information ( i ≤ p ) before it decodes its own signal. If the p -th user cannotdetect the i -th users information, outage occurs and is denotedby E p,i . Hence the outage probability of p -th user can beformulated as follows: P p =1 − Pr (cid:2) E cp, ∩ E cp, ∩ · · · ∩ E cp,p (cid:3) , (8)where E cp,i denotes the complement event of E p,i . Theorem 1.
Under the condition of ipSIC scheme, the exactexpression of outage probability for the p -th terrestrial userin NOMA-based satellite network is given by P pipSIC = Θ p Ω I M − p X l =0 (cid:18) M − pl (cid:19) ( − l α p + lp p + l Z ∞ " ∞ X k =0 ( m p ) k ( k !) × δ kp β k +1 p γ (cid:0) k + 1 , ψ ∗ p ( ηρx + 1) β p (cid:1) p + l e − x Ω I dx, (9) where Θ p = M !( M − p )!( p − , ψ ∗ p = max { ψ , ..., ψ p } , ψ p = γ thp ρφ p a p − γ thp M P i = p +1 a i ! with a p > γ th p M P i = p +1 a i , ψ M = γ thM ρφ M a M , γ th p = 2 ˜ R p − with ˜ R p being the target data rateat the p -th user to detect x p . It is worth noting that the first user (i.e., p = 1 ) with the worsechannel condition does not perform SIC operation. Hence there is no residualinterference term ηρ | h I | in (6). Proof.
On the basis of (5), (6), and (7), (8) can be calculatedas follows: P pipSIC = 1 − Pr h | h p | > ψ ∗ p (cid:16) ηρ | h I | + 1 (cid:17)i = 1 − Z ∞ Z ∞ ψ ∗ p ( ̟ρx +1) f | h I | ( x ) f | h p | ( y ) dydx = Z ∞ F | h p | (cid:0) ψ ∗ p ( ηρx + 1) (cid:1) I e − x Ω I dx. (10)Based on [17], the relationship of CDF between the orderedchannel gain and unordered channel gain can be expressed as F | h p | ( x ) = M !( p − M − p )! M − p X l =0 (cid:18) M − pl (cid:19) × ( − l p + l (cid:16) F | ˆ h p | ( x ) (cid:17) p + l , (11)where F | ˆ h p | ( x ) is the CDF of unsorted channel gain. Uponsubstituting (1) into (11) and combining (10), we can obtain(9). The proof is completed. Corollary 1.
Under the condition of pSIC scheme, the closed-form expression of outage probability for the p -th user inNOMA-based satellite network can be given by P ppSIC =Θ p M − p X l =0 (cid:18) M − pl (cid:19) ( − l α p + lp p + l × " ∞ X k =0 ( m p ) k δ kp ( k !) β k +1 p γ (cid:0) k + 1 , ψ ∗ p β p (cid:1) p + l . (12)
2) Diversity order:
To get more insights, the diversity orderis usually selected to be a metric, which is capable of describ-ing how fast outage probability decreases with the transmitSNRs [2, 21]. Based on these explanations, the diversity orderof terrestrial user can be given by d = − lim ρ →∞ log ( P ∞ ( ρ ))log ρ , (13)where P ∞ ( ρ ) denotes the asymptotic outage probability. Corollary 2.
The asymptotic outage probability of the p -thterrestrial user with ipSIC in the high SNR regime is given by P p, ∞ ipSIC = Θ p Ω I M − p X l =0 (cid:18) M − pl (cid:19) ( − l α p + lp p + l Z ∞ " ∞ X k =0 ( m p ) k ( k !) × δ kp β k +1 p γ (cid:0) k + 1 , ηxϑ ∗ p β p (cid:1) p + l e − x Ω I dx, (14) where ϑ ∗ p = max { ϑ , . . . , ϑ p } and ϑ p = γ thp φ p a p − γ thp M P i = q +1 a i ! .Proof. We commence the diversity order analyses by char-acterizing the CDF P pipSIC in the high SNR regime. When ρ → ∞ , the terms ψ ∗ p and ρψ ∗ p of P pipSIC are equal to zeroand ρϑ ∗ p , respectively. Upon substituting these terms into (9),we can obtain (14). Noting that P p, ∞ ipSIC is a constant valuewith increasing the SNRs. The proof is completed. Remark 1.
Upon substituting (14) into (13) , the p -th terres-trial user with ipSIC achieves the zero diversity order. This isdue to the influence of residual interference from ipSIC. Corollary 3.
The asymptotic outage probability of the p -thterrestrial user with pSIC in the high SNR regime is given by P p, ∞ pSIC = M !( M − p )! p ! α pp (cid:0) ψ ∗ p (cid:1) p ∝ ρ p , (15) where ∝ denotes “be proportional to”.Proof. By invoking series representation [19, Eq. (8.354.1)],the term γ (cid:0) k + 1 , ψ ∗ p β p (cid:1) of P ppSIC can be further written as γ (cid:0) k + 1 , ψ ∗ p β p (cid:1) = ∞ P n =0 ( − n ( ψ ∗ p β p ) k +1+ n n !( k +1+ n ) . When ρ → ∞ ,that is ψ ∗ p → and taking the first term ( n = 0) of seriesrepresentation, the asymptotic analysis of γ (cid:0) k + 1 , ψ ∗ p β p (cid:1) isgiven by γ (cid:0) k + 1 , ψ ∗ p β p (cid:1) ≈ (cid:0) ψ ∗ p β p (cid:1) k +1 k + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ∗ p → . (16)Upon substituting (16) into (12), the outage probability P ppSIC can be approximated as P ppSIC ≈ Θ p M − p X l =0 (cid:18) M − pl (cid:19) ( − l α p + lp p + l × ∞ X k =0 ( m p ) k δ kp (cid:0) ψ ∗ p (cid:1) k +1 ( k !) ( k + 1) ! p + l . (17)Based on [16, Eq. (26)] and further taking the first term ofseries representation in (17), i.e., k = 0 and l = 0 , we canobtain (15). The proof is completed. Remark 2.
Upon substituting (15) into (13) , the diversityorder of p -th terrestrial user with pSIC is equal to p , whichis closely related to the order of channel gains. IV. N
UMERICAL R ESULTS
In this section, the numerical results are provided andshow the impact of system parameters on NOMA-basedsatellite communication network. The links between satelliteand terrestrial users are subject to Shadowed-Rician fadingwith channel parameters given in Table I [18]. Monte Carlosimulation parameters used in this section are summarized inTable II [22]. We assume that there are three users in thenetwork, i.e., M = 3 . The power allocation factors for multipleusers are set to be a = 0 . , a = 0 . , a = 0 . , respectively.Without loss of generality, the conventional OMA is selectedto be a baseline, where the target rate R o of orthogonal useris equal to the sum rate of non-orthogonal users, R = 0 . , R = 0 . and R = 1 bit per channel use.Fig. 1 plots the outage probability versus the transmitSNR with satellite channel experiencing FHS. The exactoutage probability of curves for the p -th terrestrial user (i.e., p = 1 , p = 2 and p = 3 ) with ipSIC/pSIC are given bynumerical simulations and perfectly match with the analyticalexpressions. The asymptotic curves well approximate the exactoutage probability curves. Due to the influence of residual TABLE I: Table of Parameters for Satellite CommunicationsChannel
Shadowing b m Ω Frequent heavy shadowing (FHS) 0.063 0.739 . × − Average shadowing (AS) 0.126 10.1 0.835Infrequent light shadowing (ILS) 0.158 19.4 1.29
TABLE II: Table of Parameters for Numerical Results
Monte Carlo simulations repeated iterationsSatellite orbit type LEOCarrier frequency GHz3dB angle ϕ . ◦ User’s antenna gain per beam . dBiSatellite’s antenna gain per beam . dBiThe distance between satellite and users kmThe angle between the beam center and users . ◦ −6 −5 −4 −3 −2 −1 SNR (dB) O u t age P r obab ili t y SimulationError floorAsymptoticConventional OMAUser 1 − Exact analysisUser 2 − Exact analysis − pSICUser 2 − Exact analysis − ipSICUser 3 − Exact analysis − pSICUser 3 − Exact analysis − ipSIC
E{|h I | }=−27 dBE{|h I | }=−20 dBE{|h I | }=−30 dB Fig. 1: Outage probability versus the transmit SNR.interference, the outage probability of terrestrial users withipSIC converge to an error floor. With increasing the value ofresidual interference, the outage behaviors of terrestrial user( p = 2 ) with ipSIC are getting worse compared to other users.Another observation is that the outage performance of non-orthogonal users with pSIC is superior to that of orthogonaluser. The basic reason for this phenomenon is that NOMAis capable of providing much more fairness when it servesmultiple users at the same time [2].Fig. 2 plots the outage probability versus the transmit SNRwith different satellite channel parameters for the simulationsetting ϕ = 0 . ◦ , ϕ = 0 . ◦ , ϕ = 0 . ◦ . We observe thatthe outage behaviors of users are sensitive to the shadowingcondition of satellite-terrestrial channels. It is shown thatthe shadowing degrades network performance significantly.Frequent heavy shadowing results in a increasing outageperformance, since the higher shadowing severities correspondto worse propagation conditions. As the value of channelshadowing parameter, i.e., b , m , and Ω decreases, the outageperformance of terrestrial users is becoming much worseseriously. This is due to the fact that both LoS componentand multipath component become smaller for NOMA-basedsatellite network.Fig. 3 plots the outage probability versus the transmit SNR −6 −4 −2 SNR (dB) O u t age P r obab ili t y SimulationAsymptoticUser 1 − Exact analysis − pSICUser 2 − Exact analysis − pSICUser 3 − Exact analysis − pSIC
FHSASILS
Fig. 2: Outage probability versus the transmit SNR. −6 −4 −2 SNR (dB) O u t age P r obab ili t y SimulationAsymptoticUser 1 − Exact analysis − pSICUser 2 − Exact analysis − pSICUser 3 − Exact analysis − pSIC φ = 0.6 ° , φ = 0.5 ° , φ = 0.4 °φ = φ = φ = 0.1 ° Fig. 3: Outage probability versus the transmit SNR.with different angles between the beam center and users forexperiencing FHS. One can observe from figure that with theangles increasing, the outage behaviors of terrestrial users arebecoming much worse. This is due to the fact that with theincrease of angles, the users are getting closer to the edgeof the beam relative to satellite. As a result, to obtain bettersystem performance, we should adjust the angle of satelliteto target the terrestrial users from the perspective of servicequality. V. C
ONCLUSION
In this paper, the application of NOMA to satellite com-munication network has been investigated over Shadowed-Rician Fading Channels. The impact of system parameterson the performance of NOMA-based satellite network hasbeen discussed, where the terrestrial users with ipSIC/pSIC areconsidered carefully. New exact and asymptotic expressionsof outage probability for terrestrial users have been derivedto characterize the network performance. Simulation resultshave shown that the outage behaviors of NOMA-based satellitenetwork with pSIC is superior to that of OMA. R EFERENCES[1] Y. Liu, Z. Qin, M. Elkashlan, Z. Ding, A. Nallanathan, and L. Hanzo,“Non-orthogonal multiple access for 5G and beyond,”
Proceedings ofthe IEEE , vol. 105, no. 12, pp. 2347–2381, Dec. 2017.[2] Z. Ding, Z. Yang, P. Fan, and H. V. Poor, “On the performance ofnon-orthogonal multiple access in 5G systems with randomly deployedusers,”
IEEE Signal Process. Lett. , vol. 21, no. 12, pp. 1501–1505, Dec.2014.[3] J. Choi, “Non-orthogonal multiple access in downlink coordinated two-point systems,”
IEEE Commun. Lett. , vol. 18, no. 2, pp. 313–316, Feb.2014.[4] Z. Ding, M. Peng, and H. V. Poor, “Cooperative non-orthogonal multipleaccess in 5G systems,”
IEEE Commun. Lett. , vol. 19, no. 8, pp. 1462–1465, Aug. 2015.[5] Y. Liu, Z. Ding, M. Elkashlan, and H. V. Poor, “Cooperative non-orthogonal multiple access with simultaneous wireless information andpower transfer,”
IEEE J. Sel. Areas Commun. , vol. 34, no. 4, pp. 938–953, Apr. 2016.[6] X. Yue, Y. Liu, S. Kang, A. Nallanathan, and Z. Ding, “Exploitingfull/half-duplex user relaying in NOMA systems,”
IEEE Trans. Com-mun. , vol. 66, no. 2, pp. 560–575, Feb. 2018.[7] Y. Liu, Z. Qin, M. Elkashlan, Y. Gao, and L. Hanzo, “Enhancing thephysical layer security of non-orthogonal multiple access in large-scalenetworks,”
IEEE Trans. Wireless Commun. , vol. 16, no. 3, pp. 1656–1672, Mar. 2017.[8] N. Zhao, D. Li, M. Liu, Y. Cao, Y. Chen, Z. Ding, and X. Wang,“Secure transmission via joint precoding optimization for downlinkMISO NOMA,”
IEEE Trans. Veh. Technol. , vol. 68, no. 8, pp. 7603–7615, Aug. 2019.[9] H. Lei, Z. Yang, K. Park, I. S. Ansari, Y. Guo, G. Pan, and M. Alouini,“Secrecy outage analysis for cooperative NOMA systems with relayselection schemes,”
IEEE Trans. Commun. , vol. 67, no. 9, pp. 6282–6298, Sep. 2019.[10] Z. Ding, P. Fan, G. K. Karagiannidis, R. Schober, and H. V. Poor,“NOMA assisted wireless caching: Strategies and performance analysis,”
IEEE Trans. Commun. , vol. 66, no. 10, pp. 4854–4876, Oct. 2018.[11] Y. Liu, Z. Qin, Y. Cai, Y. Gao, G. Y. Li, and A. Nallanathan, “UAV com-munications based on non-orthogonal multiple access,”
IEEE WirelessCommun. , vol. 26, no. 1, pp. 52–57, Feb. 2019.[12] N. Zhao, X. Pang, Z. Li, Y. Chen, F. Li, Z. Ding, and M. Alouini,“Joint trajectory and precoding optimization for UAV-assisted NOMAnetworks,”
IEEE Trans. Commun. , vol. 67, no. 5, pp. 3723–3735, May2019.[13] S. Cioni, R. D. Gaudenzi, O. D. R. Herrero, and N. Girault, “On thesatellite role in the era of 5G massive machine type communications,”
IEEE Network , vol. 32, no. 5, pp. 54–61, Sep. 2018.[14] X. Zhang, L. Zhu, T. Li, Y. Xia, and W. Zhuang, “Multiple-user transmis-sion in space information networks: Architecture and key techniques,”
IEEE Wireless Commun. , vol. 26, no. 2, pp. 17–23, Apr. 2019.[15] X. Yan, H. Xiao, C. Wang, K. An, A. T. Chronopoulos, and G. Zheng,“Performance analysis of NOMA-based land mobile satellite networks,”
IEEE Access , vol. 6, pp. 31 327–31 339, Jun. 2018.[16] X. Yue, Z. Qin, Y. Liu, S. Kang, and Y. Chen, “A unified framework fornon-orthogonal multiple access,”
IEEE Trans. Commun. , vol. 66, no. 11,pp. 5346–5359, Nov. 2018.[17] H. A. David and H. N. Nagaraja,
Order Statistics , 3rd ed. New York:John Wiley, 2003.[18] A. Abdi, W. C. Lau, M. . Alouini, and M. Kaveh, “A new simple modelfor land mobile satellite channels: first- and second-order statistics,”
IEEE Trans. Wireless Commun. , vol. 2, no. 3, pp. 519–528, May 2003.[19] I. S. Gradshteyn and I. M. Ryzhik,
Table of Integrals, Series andProducts , 7th ed. New York, NY, USA: Academic Press, 2007.[20] G. Zheng, S. Chatzinotas, and B. Ottersten, “Generic optimization oflinear precoding in multibeam satellite systems,”
IEEE Trans. WirelessCommun. , vol. 11, no. 6, pp. 2308–2320, Jun. 2012.[21] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversityin wireless networks: Efficient protocols and outage behavior,”
IEEETrans. Inf. Theory , vol. 50, no. 12, pp. 3062–3080, Dec. 2004.[22] M. R. Bhatnagar and A. M.K., “Performance analysis of af basedhybrid satellite-terrestrial cooperative network over generalized fadingchannels,”